1. 1 Nell Dale Chapter 9 Trees Plus Modified from the slides by Sylvia Sorkin, Community College of Baltimore County - Essex Campus C++ Plus Data Structures

2. 2 A special kind of binary tree in which: 1. Each leaf node contains a single operand, 2. Each nonleaf node contains a single binary operator, and 3. The left and right subtrees of an operator node represent subexpressions that must be evaluated before applying the operator at the root of the subtree. A Binary Expression Tree is...

3. 3 A Two-Level Binary Expression ‘-’ ‘8’ ‘5’ treePtr INORDER TRAVERSAL : 8 - 5 has value 3 PREORDER TRAVERSAL: - 8 5 POSTORDER TRAVERSAL: 8 5 -

4. 4 Levels Indicate Precedence When a binary expression tree is used to represent an expression, the levels of the nodes in the tree indicate their relative precedence of evaluation. Operations at higher levels of the tree are evaluated later than those below them. The operation at the root is always the last operation performed.

5. 5 A Binary Expression Tree ‘*’ ‘+’ ‘4’ ‘3’ ‘2’ What value does it have? ( 4 + 2 ) * 3 = 18

6. 6 A Binary Expression Tree ‘*’ ‘+’ ‘4’ ‘3’ ‘2’ What infix, prefix, postfix expressions does it represent?

7. 7 A Binary Expression Tree ‘*’ ‘+’ ‘4’ ‘3’ ‘2’ Infix: ( ( 4 + 2 ) * 3 ) Prefix: * + 4 2 3 Postfix: 4 2 + 3 * has operators in order used

8. 8 Inorder Traversal: (A + H) / (M - Y) ‘/’ ‘+’ ‘A’ ‘H’ ‘-’ ‘M’‘Y’ tree Print left subtree firstPrint right subtree last Print second

9. 9 Preorder Traversal: / + A H - M Y ‘/’ ‘+’ ‘A’ ‘H’ ‘-’ ‘M’‘Y’ tree Print left subtree secondPrint right subtree last Print first

10. 10 ‘/’ ‘+’ ‘A’ ‘H’ ‘-’ ‘M’‘Y’ tree Print left subtree firstPrint right subtree second Print last Postorder Traversal: A H + M Y - /

11. 11 Evaluate this binary expression tree ‘*’ ‘-’ ‘8’ ‘5’ What infix, prefix, postfix expressions does it represent? ‘/’ ‘+’ ‘4’ ‘3’ ‘2’

12. 12 A binary expression tree Infix: ( ( 8 - 5 ) * ( ( 4 + 2 ) / 3 ) ) Prefix: * - 8 5 / + 4 2 3 Postfix: 8 5 - 4 2 + 3 / * has operators in order used ‘*’ ‘-’ ‘8’ ‘5’ ‘/’ ‘+’ ‘4’ ‘3’ ‘2’

13. 13 ExprTreeNode (Lab 11) class ExprTreeNode { private: ExprTreeNode (char elem, ExprTreeNode *leftPtr, ExprTreeNode *rightPtr); // Constructor char element; // Expression tree element ExprTreeNode *left, // Pointer to the left child *right; // Pointer to the right child friend class Exprtree; };. left. element. right NULL 6000 *

14. 14 InfoNode has 2 forms enum OpType { OPERATOR, OPERAND } ; struct InfoNode { OpType whichType; union// ANONYMOUS union { char operation ; int operand ; } };. whichType. operation OPERATOR ‘+’. whichType. operand OPERAND 7

15. 15 Each node contains two pointers struct TreeNode { InfoNode info ; // Data member TreeNode* left ; // Pointer to left child TreeNode* right ; // Pointer to right child };. left. info. right NULL 6000. whichType. operand OPERAND 7

16. 16 Function Eval() Definition: Evaluates the expression represented by the binary tree. Size: The number of nodes in the tree. Base Case: If the content of the node is an operand, Func_value = the value of the operand. General Case: If the content of the node is an operator BinOperator, Func_value = Eval(left subtree) BinOperator Eval(right subtree)

17. 17 Eval(TreeNode * tree) Algorithm: IF Info(tree) is an operand Return Info(tree) ELSE SWITCH(Info(tree)) case + :Return Eval(Left(tree)) + Eval(Right(tree)) case - : Return Eval(Left(tree)) - Eval(Right(tree)) case * : Return Eval(Left(tree)) * Eval(Right(tree)) case / : Return Eval(Left(tree)) / Eval(Right(tree))

18. int Eval ( TreeNode* ptr ) // Pre: ptr is a pointer to a binary expression tree. // Post: Function value = the value of the expression represented // by the binary tree pointed to by ptr. { switch ( ptr->info.whichType ) { case OPERAND : return ptr->info.operand ; case OPERATOR : switch ( tree->info.operation ) { case ‘+’ : return ( Eval ( ptr->left ) + Eval ( ptr->right ) ) ; case ‘-’ : return ( Eval ( ptr->left ) - Eval ( ptr->right ) ) ; case ‘*’ : return ( Eval ( ptr->left ) * Eval ( ptr->right ) ) ; case ‘/’ : return ( Eval ( ptr->left ) / Eval ( ptr->right ) ) ; } 18

19. 19 class ExprTree ‘*’ ‘+’ ‘4’ ‘3’ ‘2’ ExprTree ~ExprTree Build Evaluate. private: root

20. 20 A Nonlinked Representation of Binary Trees Store a binary tree in an array in such a way that the parent-child relationships are not lost

21. 21 A full binary tree A full binary tree is a binary tree in which all the leaves are on the same level and every non leaf node has two children. SHAPE OF A FULL BINARY TREE

22. 22 A complete binary tree A complete binary tree is a binary tree that is either full or full through the next-to-last level, with the leaves on the last level as far to the left as possible. SHAPE OF A COMPLETE BINARY TREE

23. 23 What is a Heap? A heap is a binary tree that satisfies these special SHAPE and ORDER properties: n Its shape must be a complete binary tree. n For each node in the heap, the value stored in that node is greater than or equal to the value in each of its children.

24. 24 Are these both heaps? C A T treePtr 50 20 18 30 10

25. 25 70 60 40 30 12 810 tree Is this a heap?

26. 26 70 60 40 30 12 8 tree Where is the largest element in a heap always found? “maximum heap”

27. 27 70 0 60 1 40 3 30 4 12 2 8 5 tree We can number the nodes left to right by level this way

28. 28 70 0 60 1 40 3 30 4 12 2 8 5 tree And use the numbers as array indexes to store the tree [ 0 ] [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ] [ 6 ] 70 60 12 40 30 8 tree.nodes

29. 29 Parent-Child Relationship? tree.nodes[index]: left child: tree.nodes[index*2 + 1] right child: tree.nodes[index*2 + 2] parent: tree.nodes[(index-1) / 2] Leaf nodes: tree.nodes[numElements / 2] … tree.nodes[numElements - 1]

30. 30 An application … Fast access to the largest (or highest- priority) element in the structure: - remove the element with the largest value from a heap …

31. // HEAP SPECIFICATION // Assumes ItemType is either a built-in simple data type // or a class with overloaded realtional operators. template struct HeapType { void ReheapDown ( int root, int bottom ) ; void ReheapUp ( int root, int bottom ) ; ItemType* elements ; // ARRAY to be allocated dynamically int numElements ; }; 31

32. 32 ReheapDown(root, bottom) IF elements[root] is not a leaf Set maxChild to index of child with larger value IF elements[root] < elements[maxChild]) Swap(elements[root], elements[maxChild]) ReheapDown(maxChild, bottom)

33. 33 // IMPLEMENTATION OF RECURSIVE HEAP MEMBER FUNCTIONS template void HeapType ::ReheapDown ( int root, int bottom ) // Pre: root is the index of the node that may violate the heap // order property // Post: Heap order property is restored between root and bottom { int maxChild ; int rightChild ; int leftChild ; leftChild = root * 2 + 1 ; rightChild = root * 2 + 2 ; 33 ReheapDown()

34. if ( leftChild <= bottom )// ReheapDown continued { if ( leftChild == bottom ) maxChild = leftChld ; else { if (elements [ leftChild ] <= elements [ rightChild ] ) maxChild = rightChild ; else maxChild = leftChild ; } if ( elements [ root ] < elements [ maxChild ] ) { Swap ( elements [ root ], elements [ maxChild ] ) ; ReheapDown ( maxChild, bottom ) ; } 34

35. // IMPLEMENTATIONcontinued template void HeapType ::ReheapUp ( int root, int bottom ) // Pre: bottom is the index of the node that may violate the heap // order property. The order property is satisfied from root to // next-to-last node. // Post: Heap order property is restored between root and bottom { int parent ; if ( bottom > root ) { parent = ( bottom - 1 ) / 2; if ( elements [ parent ] < elements [ bottom ] ) { Swap ( elements [ parent ], elements [ bottom ] ) ; ReheapUp ( root, parent ) ; } 35

36. 36 Priority Queue A priority queue is an ADT with the property that only the highest-priority element can be accessed at any time.

37. Priority Queue ADT Specification Structure: The Priority Queue is arranged to support access to the highest priority item Operations: n MakeEmpty n Boolean IsEmpty n Boolean IsFull n Enqueue(ItemType newItem) n Dequeue(ItemType& item) 37

38. ADT Priority Queue Operations Transformers n MakeEmpty n Enqueue n Dequeue Observers n IsEmpty n IsFull change state observe state 38

39. Dequeue(ItemType& item) Function: Removes element with highest priority and returns it in item. Precondition: Queue is not empty. Postcondition: Highest priority element has been removed from queue. Item is a copy of removed element. 39

40. // CLASS PQTYPE DEFINITION AND MEMBER FUNCTIONS //-------------------------------------------------------- #include "bool.h" #include "ItemType.h" // for ItemType template class PQType { public: PQType( int ); ~PQType ( ); void MakeEmpty( ); bool IsEmpty( ) const; bool IsFull( ) const; void Enqueue( ItemType item ); void Dequeue( ItemType& item ); private: int numItems; HeapType items; int maxItems; }; 40

41. PQType ~PQType Enqueue Dequeue. class PQType [0] [1] [2] [3] [4] [5] [6] [7] [8] [9] ‘X’ ‘C’ ‘J’ Private Data: numItems 3 maxItems 10 items.elements.numElements

42. Implementation Level Algorithm: Dequeue(): O(log 2 N) n Set item to root element from queue n Move last leaf element into root position n Decrement numItems n items.ReheapDown(0, numItems-1) Enqueue(): O(log 2 N) n Increment numItems n Put newItem in next available position n items.ReheapUp(0, numItems-1) 42

43. Comparison of Priority Queue Implementations 43 EnqueueDequeue HeapO(log 2 N) Linked ListO(N)O(1) Binary Search Tree BalancedO(log 2 N) SkewedO(N) Trade-offs: read Text page 548

44. 44 End